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Fixed points of convex and generalized convex contractions

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Abstract

Istr\(\check{a}\)tescu (Lib Math 1:151–163, 1981) introduced the notion of convex contraction. He proved that each convex contraction has a unique fixed point on a complete metric space. In this paper we study fixed points of convex contraction and generalized convex contractions. We show that the assumption of continuity condition in [11] can be replaced by a relatively weaker condition of k-continuity under various settings. On this way a new and distinct solution to the open problem of Rhoades (Contemp Math 72:233–245, 1988) is found. Several examples are given to illustrate our results.

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References

  1. Alghamdi, M.A., Alnafei, S.H., Radenović, S., Shahzad, N.: Fixed point theorems for convex contraction mappings on cone metric spaces. Math. Comput. Model. 54, 2020–2026 (2011)

    Article  MathSciNet  Google Scholar 

  2. Alizadeh, S., Moradlou, F., Salimi, P.: Some fixed point results for \((\alpha,\beta ), (\psi -\phi )\)-contractive mappings. Filomat 28(3), 635–647 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bisht, R.K., Pant, R.P.: A remark on discontinuity at fixed point. J. Math. Anal. Appl. 445, 1239–1242 (2017)

    Article  MathSciNet  Google Scholar 

  4. Bisht, R.K., Rakočević, V.: Generalized Meir–Keeler type contractions and discontinuity at fixed point. Fixed Point Theory 19(1), 57–64 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bisht, R.K., Hussain, N.: A note on convex contraction mappings and discontinuity at fixed point. J. Math. Anal. 8(4), 90–96 (2017)

    MathSciNet  Google Scholar 

  6. Bryant, V.W.: A remark on a fixed point theorem for iterated mappings. Am. Math. Mon. 75(4), 399–400 (1968)

    Article  MathSciNet  Google Scholar 

  7. Ding X, Cao J, Zhao X, Alsaadi FE (2017) Mittag–Leffler synchronization of delayed fractional-order bidirectional associative memory neural networks with discontinuous activations: state feedback control and impulsive control schemes. Proc. R. Soc. A Math. Eng. Phys. Sci. https://doi.org/10.1098/rspa.2017.0322

    Article  MathSciNet  Google Scholar 

  8. Forti, M., Nistri, P.: Global convergence of neural networks with discontinuous neuron activations. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50(11), 1421–1435 (2003)

    Article  MathSciNet  Google Scholar 

  9. Ghorbanian, V., Rezapour, S., Shahzad, N.: Some ordered fixed point results and the property (P). Comput. Math. Appl. 63, 1361–1368 (2012)

    Article  MathSciNet  Google Scholar 

  10. Hussain, N., Salimi, P.: Fixed points for generalized \(\psi \)-contraction with application to intergral equations. J. Nonlinear Convex Anal. 16(4), 711–729 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Istr\(\check{a}\)tescu, V.I.: Some fixed point theorems for convex contraction mappings and convex nonexpansive mapping I. Lib. Math. 1, 151–163 (1981)

  12. Istr\(\check{a}\)tescu, V.I.: Fixed Point Theory: An Introduction. Mathematics and Its Applications, vol. 7. D. Reidel Publishing Company, Dordrecht (1981)

  13. Istr\(\check{a}\)tescu, V.I.: Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters—I. Ann. Mat. Pura Appl. 130(1), 89–104 (1982)

  14. Kannan, R.: Some results on fixed points—II. Am. Math. Mon. 76, 405–408 (1969)

    MathSciNet  MATH  Google Scholar 

  15. Latif, A., Ninsri, A., Sintunavarat, W.: The \((\alpha,\beta )\)-generalized convex contractive condition with approximate fixed point results and some consequence. Fixed Point Theory Appl. 2016, 58 (2016). https://doi.org/10.1186/s13663-016-0546-z

    Article  MathSciNet  MATH  Google Scholar 

  16. Latif, A., Sintunavarat, W., Ninsri, A.: Approximate fixed point theorems for partial generalized convex contraction in \(\alpha \)-complete metric spaces. Taiwan J. Math. 19(1), 315–333 (2015)

    Article  MathSciNet  Google Scholar 

  17. Miandaragh, M.A., Postolache, M., Rezapour, S.: Approximate fixed points of generalized convex contractions. Fixed Point Theory Appl. 2013, 255 (2013)

    Article  MathSciNet  Google Scholar 

  18. Nie, X., Zheng, W.X.: On multistability of competitive neural networks with discontinuous activation functions. In: 4th Australian Control Conference (AUCC), pp. 245–250 (2014)

  19. Nie, X., Zheng, W.X.: Multistability of neural networks with discontinuous non-monotonic Piecewise linear activation functions and time-varying delays. Neural Netw. 65, 65–79 (2015)

    Article  Google Scholar 

  20. Nie, X., Zheng, W.X.: Dynamical behaviors of multiple equilibria in competitive neural networks with discontinuous nonmonotonic piecewise linear activation functions. IEEE Trans. Cybern. 46(3), 679–693 (2016)

    Article  Google Scholar 

  21. Pant, R.P.: Discontinuity and fixed points. J. Math. Anal. Appl. 240, 284–289 (1999)

    Article  MathSciNet  Google Scholar 

  22. Pant, A., Pant, R.P.: Fixed points and continuity of contractive maps. Filomat 31(11), 3501–3506 (2017)

    Article  MathSciNet  Google Scholar 

  23. Rashid, M., Batool, I., Mehmood, N.: Discontinuous mappings at their fixed points and common fixed points with applications. J. Math. Anal. 9(1), 90–104 (2018)

    MathSciNet  Google Scholar 

  24. Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14, 121–124 (1971)

    Article  MathSciNet  Google Scholar 

  25. Rhoades, B.E.: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)

    Article  MathSciNet  Google Scholar 

  26. Rhoades, B.E.: Contractive definitions and continuity. Contemp. Math. 72, 233–245 (1988)

    Article  MathSciNet  Google Scholar 

  27. Tas, N., Ozgur, N.Y.: A new contribution to discontinuity at fixed point. arXiv:1705.03699v2 (2018 )

  28. Wu, H., Shan, C.: Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses. Appl. Math. Model. 33(6), 2564–2574 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are thankful to the learned referees for suggesting some improvements in the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Computational Sciences, National Defence Academy, Khadakwasla, Pune, 411023, India

    Ravindra K. Bisht

  2. Faculty of Sciences and Mathematics, University of Nis, Visegradska 33, Nis, 18000, Serbia

    Vladimir Rakočević

Authors
  1. Ravindra K. Bisht
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  2. Vladimir Rakočević
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Correspondence to Vladimir Rakočević.

Additional information

Vladimir Rakočević is supported By Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.

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Bisht, R.K., Rakočević, V. Fixed points of convex and generalized convex contractions. Rend. Circ. Mat. Palermo, II. Ser 69, 21–28 (2020). https://doi.org/10.1007/s12215-018-0386-2

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